Global existence and large time behaviors of the solutions to the full incompressible Navier-Stokes equations with temperature-dependent coefficients

Local well‐posedness to the 2D Cauchy problem of nonhomogeneous heat‐conducting Navier–Stokes and magnetohydrodynamic equations with vacuum at infinity

“…It should be noted that system (1.1) becomes the nonhomogeneous heat-conducting Navier-Stokes equations when there is no electromagnetic field; the mathematical results concerning the global existence of strong solutions to this model can refer, for example, to several studies. [19][20][21][22][23] Although some research results have been obtained, yet it is still unknown even for the local existence of strong solutions to 2D Cauchy problems (1.1)-(1.3) with vacuum at infinity. The aim of the present paper is to investigate the local well-posedness theory of strong solutions to problems (1.1)- (1.3).…”

“…It is worth mentioning that the compatibility condition () is also needed in Zhu and Ou 18 in order to ensure the boundedness of temperature. It should be noted that system () becomes the nonhomogeneous heat‐conducting Navier–Stokes equations when there is no electromagnetic field; the mathematical results concerning the global existence of strong solutions to this model can refer, for example, to several studies 19–23 …”

Global well-posedness to the 2D Cauchy problem of nonhomogeneous heat conducting magnetohydrodynamic equations with large initial data and vacuum

“…In contrast to (1.4), the heat conducting model (1.1) is more in line with reality but the problem becomes challenging. It should be noted that (1.1) becomes the nonhomogeneous heat conducting Navier-Stokes equations when there is no electromagnetic field, we refer the reader to [17,Chapter 2] for the detailed derivation of such system, and the mathematical results concerning the global existence of strong solutions to this model can refer for example to [9,20,23,24]. Let's turn our attention to the system (1.1).…”

The Well Posedness for Nonhomogeneous Boussinesq Equations

“…In [15], they studied an optimal control problem for the mathematical model that describes steady non-isothermal creeping flows of an incompressible fluid through a locally Lipschitz bounded domain. In [16], the initial-boundary value problem of completely incompressible Navier-Stokes equations with viscosity coefficient ν and heat conductivity κ varying with temperature by the power law of Chapman-Enskog are studied. When κ = 0, the method used in [16] is not applicable.…”

“…In [16], the initial-boundary value problem of completely incompressible Navier-Stokes equations with viscosity coefficient ν and heat conductivity κ varying with temperature by the power law of Chapman-Enskog are studied. When κ = 0, the method used in [16] is not applicable. We must seek new methods to overcome the difficulty.…”